By Stefan Keppeler

The ebook offers with semiclassical equipment for structures with spin, particularly tools related to hint formulae and torus quantisation and their purposes within the idea of quantum chaos, e.g. the characterisation of spectral correlations. The theoretical instruments built right here not just have rapid purposes within the idea of quantum chaos - that is the second one concentration of the booklet - but additionally in atomic and mesoscopic physics. hence the intuitive realizing of semiclassical spin dynamics can also be priceless in rising matters like spintronics and quantum computation.

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**Spinning Particles - Semiclassics and Spectral Statistics**

The e-book bargains with semiclassical tools for platforms with spin, specifically tools related to hint formulae and torus quantisation and their functions within the idea of quantum chaos, e. g. the characterisation of spectral correlations. The theoretical instruments constructed right here not just have speedy purposes within the concept of quantum chaos - that's the second one concentration of the ebook - but additionally in atomic and mesoscopic physics.

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**Example text**

166) in ﬁrst order perturbation theory. 166) as it stands has a purely continuous spectrum, see [47, 48, 49, 50] and references therein. 6 Examples 51 Hamiltonian can be used in order to calculate some relativistic corrections to the eigenvalues of the non-relativistic hydrogen atom. 166) has to be treated with caution. e. from the Schr¨odinger equation of the hydrogen atom, which by Poisson summation gives rise to an exact trace formula. Then we show how to modify this trace formula along the lines discussed in the preceeding sections, in order to account for the spin contribution.

D . 111) ωk Consequently for some time t we have ωt = 2πm , m ∈ Zd \ {0} . 112) We will label all quantities on the corresponding torus by m, the vector of winding numbers. 112) we denote by Tm := 2π mj ωj ∀ j = 1, . . 113) and for the actions which characterise the respective torus we write I m . 67) deriving from a rational torus with winding numbers m by Am . Such a contribution can be calculated directly, as we will demonstrate below. e. 112) can be fulﬁlled. Otherwise our labeling would not be unique.

21). g. e. 124) cf. 115). 21). 2 Spin–Orbit Coupling in 2 Dimensions – σp-Billiards We will introduce a class of systems which will be used in order to illustrate various aspects of semiclassics and spectral statistics for systems with spin. In the study of quantum chaos for spinless particles so-called billiard systems are useful and popular examples. In this context a classical twodimensional Euclidean billiard is deﬁned by a compact domain D ⊂ R2 with boundary ∂D. Inside the billiard one considers the motion of a free particle with elastic reﬂections at the boundary.